On smooth curves passing through rational surface singularities
type: Item
instance of: scholarly article
title: On smooth curves passing through rational surface singularities
zbMATH Open document ID: 0834.14020
DOI: 10.1007/BF02571910
author: Marco Manetti
published in: Mathematische Zeitschrift
publication date: 1995-07-26T00:00:00Z
full work available at URL: https://eudml.org/doc/174735
review text: Let \((X,C)\) be a pair where \(X\) is a germ of a normal two-dimensional singularity and \(C \subset X\) is a germ of smooth curve. Jaffe gave a complete description of such pairs up to isomorphisms when \(X\) is a rational double point in terms of the associated Dynkin diagram. In this paper that result is extended to a more general class of rational singularities. The description of the set of isomorphism classes of \((X,C)\) is in terms of the points that the fundamental cycle \(E\) in the minimal resolution of the singularity cuts out on the strict transform of \(C\) (via the resolution map). The result relies on some properties pointed out in a thorough study of the minimal resolution of the singularity.
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 778951
zbMATH Keywords: permissible point
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: rational singularities
zbMATH Keywords: minimal resolution
reviewed by: Marino Palleschi
MaRDI profile type: Publication
cites work: On Isolated Rational Singularities of Surfaces
cites work: Rationale Singularitäten komplexer Flächen
cites work: Local contributions to global deformations of surfaces
cites work: Local geometry of smooth curves passing through rational double points
cites work: Taut two-dimensional singularities
cites work: Riemann-Roch and smoothings of singularities
OpenAlex ID: W2007530960
description: scientific article; zbMATH DE number 778951